Conditional Hardness for Approximate Coloring

We study the approximate-coloring(q,Q) problem: Given a graph G, decidewhether \chi(G) \le q or \chi(G)\ge Q. We derive conditionalhardness for this problem for any constant 3\le q < Q. For q \ge4, our result is based on Khot's 2-to-1 conjecture [Khot'02].For q=3, we base our hardness result on a certain `alpha-shaped' variant of his conjecture.

We also prove that the problem almost-approximate-3-coloring(eps) is hard for any constant eps>0, assuming Khot'sUnique Games conjecture. This is the problem of deciding for a given graph, between the case whereone can 3-color all but an \eps fraction of the vertices withoutmonochromatic edges, and the case where the graph contains noindependent set of relative size at least \eps.

Our result is based on bounding various generalized noise-stabilityquantities using the invariance principle of Mossel et al [MOO'05].